The Basic Physics of the Binary Black Hole Merger GW150914

Home , Binary black hole, Chirp mass

Original Paper 1 = (1) G (1 of 17) 1600209 , km , called the Schwarzschild m m M kg is the mass of the Sun, s is the speed of light. According 95 30 . / 2 16 [9, 10]. 10 m + = 8 kg is Newton’s gravitational constant, × 2 s 10 2 / 99 c . 3 Gm × 1 2 m = = 11 998 . ) − 2 m ( 10 = × c and reproduction in any medium, provided the original work is properly cited. This is an open access article under the termsCommons of Attribution the License, Creative which permits use, distribution Here, the result that GW150914 was emitted by the in- Specifically, while the orbital motion of two bodies In the terminology of GR corrections to Newtonian dynamics,(4) (3) constitute & the “0th post-Newtonian” approximation (0PN) (see Sec. 4.4). A similar approximation was used for the firstbinary analysis pulsar of PSR 1913 Full author list appears at the end. lvc.publications@ligo.org 67 . Schwarz hole [8]. Once the blackcomes within hole this is radius can formed, no any longer object escape out that of it. spiral and merger of two blackstrain holes data follows from visible (1) at the thesional instrument and output, scaling (2) arguments, dimen- orbital (3) dynamics primarily and Newtonian (4)mula for the the Einstein luminosity quadrupole of a for- gravitational wave source. to the hoop conjecture, ifpressed a to non-spinning within mass that is radius, then com- it must form a black radiation can escape. There is a naturaldius” “gravitational associated ra- with a mass radius, given by 1 These calculations are straightforward enough thatcan they be readily verified withtime. Our pencil presentation is and by design paper approximate, empha- insizing a simple arguments. short is approximated by Newtonianlaws dynamics to and high Kepler’s precisionand sufficiently at low velocities, sufficiently we will largedynamics invoke separations Newtonian to describe thepoint motion of even orbital toward motion the end (We revisit this assumption in ∗ ∗∗ 6 and where M r ∗∗ , ∗ published by Wiley-VCH Verlag GmbH & Co. KGaA Weinheim , still orbited each other as close as , No. 1–2, 1600209 (2017) / DOI 10.1002/andp.201600209 35 M Annalen der Physik 529

km apart and subsequently merged to form a single

A black hole is a region of space-time where the 350 2016 The Authors. C waves. black hole. Similar reasoning, directly from the data,to is roughly estimate used how far these black holesthe were Earth, from and the energy that they radiated in gravitational ∼ gravitational field is so intense that neither matter nor black holes (BHs) orbiting around onemerging another to and form then another black hole. detector data and some basic physicsgeneral [7], physics accessible audience, to as a well asers. students This and simple teach- analysis indicates that the source is two tailed analytical and computationaland methods references (see therein [2–6] for details).be learned However, much about can the source by direct inspection of the by Einstein’s theory ofnal general was relativity clearly (GR). seen Thein by the sig- Hanford, two WA LIGO andinformation detectors Livingston, about located LA. the Extracting source of the the full signal requires de- Advanced LIGO made thetational first wave observation (GW) ofber signal, 14th, a GW150914 2015, gravi- a [1], successful on confirmation of Septem- a prediction 1 Introduction quent merger of two black holes. The blackof holes approximately were each general-relativistic analyses published elsewhere, in show- ing that the signal was produced by the inspiral and subse- concepts from Newtonian physics and general relativity, accessible to anyone with a general physics background.simple The analysis presented here is consistent with the fully strong enough to be apparent, without using anyform wave- model, in the filtered detectorstrain data.tures Here, of fea- the signal visible in the data are analyzed using The first direct gravitational-wavedetection was by made the Advanced Laser Interferometer Gravitational Wave Ob- servatory on September 14, 2015. The GW150914 signal was

Received 5 August 2016, revised 21 SeptemberPublished 2016, online accepted 22 4 September October 2016 2016 The basic physics of the binaryLIGO black Scientific and hole VIRGO Collaborations merger GW150914 Ann. Phys. (Berlin) Original Paper ietyfo h tandt nFg ymauigthe measuring by 1 obtained Fig. differences in be time data also strain the can from version evolution directly approximate An time-frequency 2. the Fig. of in depicted instrumental of is behavior signal known time-frequency the The around [14]. filter frequencies noise band-reject Hz), (35–350 a band frequency and sensitive LIGO the to filter asst ewl bv h evetkonneutron known both heaviest constrain the above and well and be made, examine to we assumptions masses 4 Sec. the In holes. justify black being with consistent small, the only and how heavy assumptions, be must simplest dis- constituents the then binary using We 3, inspiral. Sec. binary in a cuss as system for the relevant strain quantities analyzing the the determine off they read how and signal data, the of properties the describes atojcso ihms Ap )a ela htone what C). (App. as peak well the after as waveform B) the from (App. learn mass might high of objects com- astrophysical pact discuss and A), (App. power radiated and strain radiation appendices gravitational The of calculation system. a the provide of calcu- luminosity and total source, lumi- the the to lates wave distance gravitational the peak estimate to the nosity uses 5 Section stars. 2o 7 1600209 17) of (2 3 2 instrumentally data the 1: strain Fig. in observed shown is point starting Our data observed the Analyzing 2 detectors. the by output data the with begins [2–6]. elsewhere published been already have the system, describing numbers precise rigor- as fully well as The arguments, ous treatments. con- advanced results more gives with but elementary, sistent design by is presenta- here Our tion checks. straightforward, but intuition rough, build to using tool a as of and physics signals, wave the gravitational to introduction pedagogical a as intended 4.4). - Sec. in evolution shown (as its argument quantifiable in our for a late enough late quite using until well perturbation described analytic be can Einstein’sdynamics Newtonian have from departures system’s of [11–13] binary a (NR) that shown relativity solutions numerical using however, equations unreliable; non- fully analysis a wholly Newtonian any is make relativity could which general theory, of linear theory The 4.4). Sec. oe n h rewvfr’ intastosaedfiuttopin- difficult are transitions sign waveform’s true the and lower orsletecosn at crossing the resolve To references. its and [1] see work, detectors how the of details For waves. gravitational passing by caused strain the measure to interferometry laser use detectors LIGO advanced The h ae sognzda olw:orpresentation our follows: as organized is paper The is physics, basic using here, presented approach The t ewe ucsiezero-crossings successive between h t ( ∼ t www.ann-phys.org ,atrapyn band-pass a applying after ), 0 . 35 ,we h inlapiueis amplitude signal the when s, C 2 06TeAuthors. The 2016 eto 2 Section 3 nae e Physik der Annalen e 4 2015. Septem- 14, on ber (UTC) Time shown are Universal Times Coordinated 09:50:45 ms inverted. and to relative 6.9 has by strain intime Hanford back The shifted Both been notch-filtered. [1]. of and 1 bandpass- Figure been in shown have as (red), detector Hanford and (blue) 1 Figure n estimating and frequency signal in increase the time. over traced where be can [1]), (“chirp”) from (taken plot 2 Figure aeommdl epo the plot We model. waveform hw tlategtoclain,w nwta mass a that know we oscillations, eight clearly least waveform at the shows Since [15]. masses accelerating by 0 around for lasts signal Hz. the 200 of part above visible is entire indicate The frequency [1]) instantaneous final time-of-flight-delay the ms that 6.9 ac- a after stabilize. for detectors, both counting to (in cycles appears visible clearly frequency amplitude last The the the s, and 0.42 rapidly, around drops time frequency a the After thus increasing. decreasing, is is grav- period the region wave this In itational s. 0.30 mark starting time the increasing, around from initially is amplitude whose pattern physical its explain below. and 3, relevance Fig. in frequencies estimated .P bo ta. h ai hsc ftebnr lc oemre GW150914 merger hole black binary the of physics basic The al.: et Abbot P. B. (over zero-crossings adjacent five the of positions the averaged we point, ngnrlrltvt,gaiainlwvsaeproduced are waves gravitational relativity, general In wave a of cycles several by dominated is signal The ∼ h ntuetlsri aai h iigtndetector Livingston the in data strain instrumental The ersnaino h tandt satime-frequency a as strain-data the of representation A 6 ms). ulse yWlyVHVra mH&C.Ka Weinheim KGaA Co. & GmbH Verlag Wiley-VCH by published f GW = 1 / (2 t ,wtotasmn a assuming without ), − 8 / oe fthese of power 3 . 15s. Original Paper (2) (4) (3) (3 of 17) 1600209 . : Einstein found [17] that at a (luminosity) distance 75 Hz h × at leading order in the veloc- π A 2 x A = that of the orbital frequency (for a about the orbital axis, the radiation m A π max , www.ann-phys.org , twice 2 : The single most important quantity for ij GW 2 f t Q max π 2 d (defined in App. A) is 2 d 150 Hz ij L GW = f ∼ d Q G 4 2 c max max Determining the frequency at maximum strain am- The eight gravitational wave cycles of increasing fre- Determining the mass scale = from a system whose traceless mass quadrupole mo- Kep ij L GW f h where here and elsewhere the notation indicatesquantity that the before the vertical lineindicated is after evaluated the at the line.tional time We thus data interpret as the indicatingeach observa- that other the (roughly Keplerian bodiesan dynamics) were orbital up angular orbiting frequency to at least ω evolution is that the system consistedthat of orbited two each black other holes and subsequently merged. plitude ity) whose first time derivative is thetum, total which linear is momen- conserved for asecond closed derivative system, therefore and vanishes. whose Hence,order, gravitational at radiation is leading quadrupolar. Because the quadrupole moment (defined inunder rotations App. by A) ishas symmetric a frequency detailed calculation for a 2-bodypp. system, 356-357 see of App. [16]). A and quency therefore require at least four orbitalat revolutions, separations large enough (compared tobodies) the size that of the the bodiesquency signal do eventually terminates, not suggesting collide. theof inspiraling end The orbital motion. rising As the fre- amplitudeand decreases the frequency stabilizes theble system returns equilibrium to a configuration. sta- Weonly reasonable shall explanation show for the that observed the frequency and that thethese rate gravitational at which waves energy is is given carried by away the by quadrupole the reasoning in this paperquency is the at gravitational wave which fre- tude. the Using the waveform zero-crossings around has theand/or peak maximum the of brightest Fig. ampli- point of 1 vative Fig. (low) value 2, we take the conser- to the second time derivativement. This of is because the the electric gravitational analog dipole isdipole the mo- moment mass ( ment is the gravitational wave strain d 4 ;wehave 9 . 0 , respectively. using Eq. 8. The ∼ M H1 − 2 L1 37 M R and 40 . While this interpolation ) ∼ t . The slope of this fitted line ( 8 3 M . / 8 0 − published by Wiley-VCH Verlag GmbH & Co. KGaA Weinheim GW f ∼ of 30 2 L1 R M and 9 . 0 , No. 1–2 (2017) Annalen der Physik ∼ 529 2 H1 R A linear fit (green) of Gravitational radiation has many aspects analogous During the period when the gravitational wave fre- is not governed by gravitational waves, is explored in App.shown A.1 to and be inconsistent with this data. The possibility of a different inspiraling system, whose evolution 2016 The Authors. C to electromagnetic (EM)charges. A significant radiation difference is that from thereto is EM no accelerating dipole analog radiation, whose amplitude is proportional blue and red lines indicate 4 Figure 3 Ann. Phys. (Berlin) spiral”), increasing the orbital frequency andthe amplifying gravitational wave energy output from the system. two bodies is theonly only “damping forces” plausible are provided explanation: by gravitational there, wave emission, the which brings the orbiting bodies closer (an “in- Here, the data demonstrate very different behavior. quency and amplitude are increasing, orbital motion of since oscillations aroundcharacterized equilibrium by aredecaying roughly generically amplitudes. For example, constant inball, the the case frequencies oscillations of would a be and damped fluid by viscous forces. this time the oscillation frequencyis of increasing. the source This system initialperturbed behavior system returning cannot back be to due stable to equilibrium, a or masses are oscillating.wave The frequency and increase amplitude in also indicate gravitational that during waves of the same amplitudes and frequenciesstrain added data just to before the GW150914. LIGO A similar error estimatefound has using been the differences between H1 and zero-crossings. L1 The error-bars have been estimated by repeating the procedure for gives an estimate of the chirp mass of also found lar fit can be done usingfit either shown H1 residualhas L1 or independently.strain The squares sum of used the combined strain dataof from L1 H1 with time and shifted L1 and (in sign-flipped fact, the H1, as explained). sum A simi- Original Paper q atr4 factor a M mass, chirp the to gravi- waves emitted tational relating of 21]) derivative frequency [20, and (following frequency A simple the App. a in system, derived a is formula of luminosity wave gravitational for the formula quadrupole Einstein’s and gravitation, of law m n rqec eiaieo h rvttoa ae at waves gravitational the of derivative frequency frequency and the using data, observational the from directly approxima- is 4.4). Newtonian equation Sec. the (see This valid as [22]). is long tion of as 3 hold to Eq. expected and A5 Eq. (see quency where theintegralisoverasphereatradius 4o 7 1600209 17) of (4 5 [17] formula utb vrgdoe sy n orbit. one (say) over averaged be must ue asby mass duced m ignored is 4.6). universe Sec. the (see of expansion the gravitational and the weak where is field [19], zone” “wave the in plicable tani o o ag 1] ewl pl tutltefre- the until it apply will we [15]; quency the large and too light, not of speed is the strain orbit- to close the too of not are velocities objects ing the when waves, gravitational to msinfo iaysse,aueu asqatt is quantity mass useful the a system, binary a from emission M d E = 1 1 d qain h eta qaino eea relativity. general of equation central the Equation, Einstein the linearizing by obtained results, these for of [18] derivation of a 974-977 pp. and calculation, worked-out a for A App. See GW sn etnslw fmto,Nwo’ universal Newton’s motion, of laws Newton’s Using hs au o h hr ascnb determined be can mass chirp the for value a Thus, o h iaysse ednt h w assby masses two the denote we system binary the For norcs,E.5gvstert fls fobtlenergy orbital of loss of rate the gives 5 Eq. case, our In = = t ≥ and hr mass chirp m m c ( G 1 m f = 3 / ˙ ( 2 GW m m m f 1 GW othat so where 16 π 1 2 + 2 m h oa asby mass total the , d = c 96 π n ihu oso eeaiyasm that assume generality of loss without and m L 5 max 3 2 2 d ,adteqatt ntergthn side right-hand the on quantity the and ), ) G 2 , 3 ) / f M e e.44 hswv ecito sap- is description wave This 4.4. Sec. see , 1 3 μ 5 GW q h / ˙ π 5 ≥ = eae otecmoetmse by masses component the to related , 2 . − / d 8 = .T ecietegaiainlwave gravitational the describe To 1. m h ˙ t ( 1 i f 2 stert-fcag ftefre- the of rate-of-change the is , GW m d j 3 = www.ann-phys.org S 2 1 ) / − = d M d 11 h t 1 5 edfietems ratio mass the define We . ij M c G d f ˙ 5 d GW h = t i ij , j 3 m = , 3 1 1 5 d + 1 d 3 d / Q 5 t m L 3 , ij 2 (contributing ,andthere- d C d 3 06TeAuthors. The 2016 Q t 3 ij , (6) (7) (5) nae e Physik der Annalen hc osntinvolve not does which ofidsmlrrsls ehneot dp conserva- 3, of a estimate adopt lower henceforth tive Fig. We in results. presented similar find analysis, to an such performed have growing. stops plitude uet htw rsn ee ofrsmlct etake we simplicity for so M here, present we that guments of value exact to The 35%. constant within remains however, The value, orders-of-magnitude. mass two chirp than implied more by varies tive uso frdhf rmtesuc rm per in appears frame Dis- source frame. the detector 4.6. from Sec. the redshift in quantities given of the are and cussion it it from thus mea- derive frame, quantities we detector from the derived in is sured mass this that remark nepeain h atta h mltd ftegrav- the of amplitude the that fact The interpretation. rvttoa aefeunyabove frequency at wave down gravitational a break certainly assumptions these that period indicate a also data and The laws. Kepler’s radius by instantaneously changing described adiabatically mo- orbital an the and has light, of tion speed the to system close binary too not the are in velocities the applicable: for- these are to mulae leads which calculation the assump- into the go that that tions suggests and sup- interpretation, also this frequency ports with increases strain wave itational id aisb atro ( 5 of factor a by varies riod) pnst rvttoa aefeunisi h range the in frequencies corre- visible) 30 wave is sen- gravitational hence the to (and in detector sponds is the of signal time band inspiral The sitive cover). the journal which on during figure interval (see to curve tangents 2), drawing same by the (Fig. derivative frequency data the strain for wave and fre- gravitational plot time-frequency observed the the the for from of values estimated example, be For can quency time. in moment any osatfor constant data. observational time- the the of behavior of frequency inspection direct by obtained is system ing d ewe eocosnsi h tandt.Tecon- integration The of data. strain stant the in zero-crossings between ods calculate to used be fore f .P bo ta. h ai hsc ftebnr lc oemre GW150914 merger hole black binary the of physics basic The al.: et Abbot P. B. GW − 8 < lentvl,E.7cnb nertdt obtain to integrated be can 7 Eq. Alternatively, h atta h hr asrmisapproximately remains mass chirp the that fact The radiat- the of scale mass characteristic the that Note = / 3 ( 0M 30 f t GW ) = < (8 . 5 z vrti ie h rqec (pe- frequency the time, this Over Hz. 150 π ulse yWlyVHVra mH&C.Ka Weinheim KGaA Co. & GmbH Verlag Wiley-VCH by published f 5 GW ) 8 / < 3 5H ssrn upr o h orbital the for support strong is 150Hz G t c M c M 3 stetm fcaecne We coalescense. of time the is = M f ˙ GW 5 0M 30 / 5 1 3 ietyfo h ieperi- time the from directly ,adtefeunyderiva- frequency the and ), ( xlcty n a there- can and explicitly, M t c − sntciia otear- the to critical not is o h hr as We mass. chirp the for t ) , f GW max steam- the as , (8) Original Paper .Fora 3 would be 10 [24]. Obser- ∼ R 4 R (5 of 17) 1600209 10 , and the binary 5 × 2 10 ∼ × 3 of order 1) is further evi- R ∼ R R has 6 5. www.ann-phys.org ∼ 2and ∼ A demonstration of the scale of the orbit at minimal The fact that the Newtonian/Keplerian evolution of Radio, optical and X-ray telescopes have probed the accretion disk extending much further inside [23]. 6 Schwarzschild (red, diameter 200 km) and extremalameter Kerr 100 (blue, km). di- Note theplains, masses the here are system equal; is asWhile even identification Sec. 4.2 moreof ex- compact a rigid referencetances for frame between unequal points for measuring masses. is dis- nottion unique only in really arises relativity, with this strong complica- gravitationalKeplerian fields, regime while (of in the low compactnesstentials) and low the gravitational system’s po- center-of-massTherefore if rest-frame the system can is claimed berian to argument be should used. hold, non-compact, and the constrain the Keple- distances topact. be Thus com- the possibility of non-compactness is inconsistent with the data; see also Sec. 4.4. black hole in Cyg X-1 Figure 4 separation (black, 350 km) vs. the scale of the compact radii: system of highest known orbitaltem frequency, HM the Cancri WD (RX sys- J0806), has system of two neutronbetween stars just touching, vations of orbits around ourpresence galactic of center a supermassive indicate black the hole, named26], Sgr A* with [25, the star S2 orbiting it as close as dence that the objects are highly compact. the orbit inferred fromdown when the the separation signal is about of thehole GW150914 order radii of (compactness breaks the ratio black (9) 35 M = 2 m = 1 . We also assume m , so that the total 70 M given by .Thisisdefinedasthe 35 M R = R = . 2 published by Wiley-VCH Verlag GmbH & Co. KGaA Weinheim m M 7. 5 , the binary orbit for the stellar + . / 7 1 1 1 2 350 km 10 m ∼ = = × = 2 3 2 / . The value of the chirp mass then im- m 1 2 M , No. 1–2 (2017) ∼ Annalen der Physik m = (see App. B). 529 206 km 1 R / = max m 1 GM m 2 Kep ω 350 km value. This constrains the objects to be exceedingly In our case, the bodies of mass For comparison with other known Keplerian systems, In order to quantify the closeness of the two objects Around the time of peak amplitude the bodies there- The most compact stars known are neutron stars, = case = 2016 The Authors. C R mass would be the orbit of Mercury, thesystem, innermost has planet in our solar spinning, circular orbit, equal-mass caseR just discussed Newtonian orbital separationthe between objects the divided by centersble the of respective sum radii of (as their compact smallest objects). possi- For the non- relative to their naturalduce gravitational the radius, compactness we ratio intro- illustrated in Fig.objects, 350 4. km, The isSchwarzschild radii. orbital only about separation twice of the these sum of their each have a Schwarzschild radius of 103 km. This is masses, For simplicity, suppose that the two bodies have equal 3 Evidence for compactness in the simplest Ann. Phys. (Berlin) plies that fore had an orbital separation for now that the objectsorbits are remain not Keplerian spinning, and and essentially that circularpoint until their of the peak amplitude. is about 3 M stars could have orbited at this separation withouting collid- or merging together – butneutron star the can maximum have mass before that collapsing a into a black hole of a few mHz (far below 1 Hz). which have radii of about ten kilometers. Two neutron radii which are typically tening Eq. thousand 9 kilometers. shows Scal- that suchhave stars’ terminated inspiral with evolution a would collision at an orbital frequency before reaching suchstars close have proximity. radii measured Main-sequence millions in of hundreds kilometers, of and thousands white dwarf or (WD) stars have Compared to normaltiny length scales forsmall, stars, or else this they would is have collided a and merged long Original Paper riapsis), aiemc atrta hysrn 2,2] hscorrec- neglected. This be 21]. thus [20, can tion shrink they than faster circu- much to larize orbits the causes away carry waves gravitational monotonically. grows amplitude whose the in data, seen not is modulation and Such peaks. high-amplitude so low-amplitude between apoapsis, alternate near would than signal periapsis the near velocity greater the be [27]: modulation would a display should signal the infiatol o ihyecnrcorbits. eccentric highly for is only correction significant this However, eccentricity. the on depends tighter even is ratio orbits compactness eccentric (the by imposed bound ness 6o 7 1600209 17) of (6 7 orbital separation instan- the orbital The taneous axis. to semi-major refers the to longer rather but no separation 9) (Eq. law third Kepler’s eccentricity with orbits non-circular For eccentricity Orbital 4.1 source, effects. the potential to its distance (Sec. and the it discuss we foregoing 4.6 Sec. of In consequences 4.4–4.5). it- the assumption discuss Keplerian and the self, revisit then 4.1–4.3), modifications (Sec. three these discuss to Keple- approximation the not rian use does also We them outcome. relaxing the that change show significantly to we possible section conclusions this is in our and it affect However, assumptions these [2]). how able see examine are them, techniques constrain advanced to more are the of we (although parameters that system these possi- approximation constrain directly not of to is here, level It using the spin. at no working cir- and ble, a coalescing masses, of the equal assumptions that orbit, the show cular under to holes data black the are used objects we 3 Sec. In assumptions the Revisiting 4 R n rmblwb h on fcoetapoc (pe- approach closest of point the by below from and , eimjrai,oeobtains one axis, the semi-major and periapsis at separation the between ratio the account into be must eccentricity the increase, to ratio compactness the for Hence as(nerduigE.7 to 7) Eq. using (inferred mass cetiiyicesstelmnst 2,2]b factor a by 21] [20, luminosity the increases Eccentricity hsi o upiig steaglrmmnu that momentum angular the as surprising, not is This hr sas orcint h uioiywhich luminosity the to correction a also is There 1 e − e 2 0 r sep . − 6 7 n o atro 2, of factor a for and , / 2 ≥ 1 + 1 − 24 73 e e 2 + R etu e httecompact- the that see thus We . R 96 37 www.ann-phys.org M ssmaller). is R e r 4 sep ( ( e e e ≥ ) ) sbuddfo bv by above from bounded is = = 1 0 hsrdcn h chirp the reducing thus , . 9 1 − seFg 5) Fig. (see 3 − / 5 e ( e ) 2 · / e M 5 ( > e ( 7 ) C e ,the 0, · 06TeAuthors. The 2016 o these, For = R ( 0) e ( e ) Taking . = = R 0) of . nae e Physik der Annalen r q q ois hsgvsu ii o h aso h smaller the of mass the two for the limit object of a the us mass within gives combined This be bodies. the will of it radii that Schwarzschild compact so becomes system R nFg htbyn h asrtoof ratio mass the beyond that 5 plot minimal ratio from compactness Fig. arises the in from bound see This we data. compactness: the on purely hnoecmoe feulmasses. equal of composed is one masses than Schwarzschild unequal of frequency, their orbital composed and system of mass a chirp sum given a the for Thus, of radii. units when in smaller becomes measured objects the between separation that shows ratios clearly mass which for 5, Fig. in plotted is quantity This separation ratio compactness The hntenurnsa ii,bt oisaeepce to expected are . holes bodies black both be limit, star neutron the than rs h opnn assadttlms ntrsof ex- terms in mass we mass chirp explicitly, New- total the this and the masses see component of To the value press mass. a observed chirp implies the order for tonian that mass as total mass-ratio, ratio higher increasing compactness the with that see smaller to easy is It masses unequal of case The 4.2 χ parameter spin dimensionless the define the of mass spins a For the objects. concerns relax we assumption third The spins objects’ of effect The 4.3 h w opnn masses, component two the M Schwarz GW150914 merger hole black binary the of physics basic The al.: et Abbot P. B. ) hsalwrbudo h mle mass smaller the on bound lower a thus , 1 = = = = / n a lopaean place also can One 5 q G c m r 2( 2 Schwarz / ( 1 m m 5 π m S , + 2 2 f m 2 GW R . ,giving ), ≥ m 2 R ( 2 = 1M 11 ulse yWlyVHVra mH&C.Ka Weinheim KGaA Co. & GmbH Verlag Wiley-VCH by published M max c otesmo h cwrshl ai of radii Schwarzschild the of sum the to = q 2 M ) M G M > = (1 M n h asratio mass the and h opcns ratio compactness the 1 m (1 .Asthisis3–4timesmoremassive 2( + ) ihsi nua momentum angular spin with ω + 2 / q Kep 3 ) q 1 (1 ) / R 6 5 upper max / + q q c 5 2 − 2 q q stertoo h orbital the of ratio the is / r 3 − GM 5 ) Schwarz / 4 3 5 / / ,and 5 5 ii ntems ratio mass the on limit . ) ≈ 2 / 3 ( (1 M 3 q . + ) 0 ,as = q q ) more 2 4 r / decreases / 5 Schwarz m 5 R q 1 . m = ∼ logets also compact 2 based , M 3the 13 ( m S :the (1 1 (11) (12) (10) ) we + + Original Paper , q

= = q max (15) (14) 2 ) max c / GW f v GW ( (and f = approaches x v and (7 of 17) 1600209 3 / 2 432 M

, 1, puts a limit on the 30 M Kep 2 4 . c ω ≥ = max 3 , M R

4 M . GM 3 and a ﬁxed value of / 14 , and are enumerated by their 2 decreases as the mass ratio x =

M R ) 5 3 ) / max / 6 2 M M 2 ( ( and thus on the maximum total mass GW EK f R × q Kep r 2 www.ann-phys.org 2 c ω / 3 2 M ≤ 4 c . M . Corrections to Newtonian dynamics may G ) , from the constraint that the compactness 3 ) q G π M M 5

( 5 sep ( / / r 6 6 EK 2 2 sep 2 r c r , / 0, where dynamics are Newtonian and gravitational max The conclusion is the same as in the equal-mass or We may thus constrain the orbital compactness ratio We can also derive an upper limit on the value of the ≤ = = M max M = , these two limits coincide and may be quantified by M x GM PN order. The 0PN approximation is precisely correct at which for GW150914 implies where in the last step we used be expanded in powers of wave emission is describedformula (Eq. exactly 5). by the quadrupole – well above the neutron star mass limit (App. B). non-spinning case: both objects must be black holes. 4.4 Newtonian dynamics and compactness We now examine the applicability ofics. Newtonian dynam- The dynamics willproximation depart when from the the Newtonianthe relative ap- speed of velocity lightcomes or large when compared the to the gravitationalnary rest energy mass system be- energy. bound For a by bi- v gravity and with orbitalthe velocity post-Newtonian (PN) parameter [28] (now accounting for eccentricity,spin) unequal by masses, and R the compactness ratio increases. Thus, the constraint maximal possible 150 Hz. This3.4 constrains (1.7) the times their constituents extremalmaking Kerr to (Schwarzschild) them radii, be highly under ment compact. is illustrated in The Fig. 4. compact arrange- mass ratio 83). This again forces the smaller mass to be at least 5 M ratio must be larger than unity. This isvalue because, for of a fixed the chirp mass = = to 0 χ (13) )the max = e GW f 11 M .Asthisis 2 ∼ c is . / 2 and M m km ( Gm 13 = . The bottom-left corner ) 30 M ∼ M 8 . M m q 0 ( = ) = 5 M . ( M e published by Wiley-VCH Verlag GmbH & Co. KGaA Weinheim 1 Schwarz ≈ r modify their gravitational radii 2 1 2 Schwarz 2 , which is at most a factor of two r c = m , as explained in Sec. 4.1. The bottom edge GM 1 2 ) 27 m . = = ( 0 and Schwarz ) , No. 1–2 (2017) r 2 EK 1 Annalen der Physik r = m 529 m ( e ) beyond mass ratio of ) corresponds to the case given in Sec. 3. At fixed 0 1), 0 EK r = = = This figure shows compactnessthe constraints ratio im- + e e χ . It plots the compactness ratio (the ratio of the separa- ) , ) illustrates the argument given in Sec. 4.2 and Eq. 11: the sys- 1 1 0 m rather than The smallest radius a non-spinning object ( ( = = 2016 The Authors. EK EK e q C Figure 5 posed on the binary system by Ann. Phys. (Berlin) a lower limit on thecent non-black Newtonian hole separation bodies of of total two mass adja- linear in the mass, and summing radii linearly, we obtain larger than for non-spinning objects. nent Schwarzschild radii. The spins of system would become more compact than the sum of the compo- of two, to the radiuswhich of an extremal Kerr black hole (for 0) couldSchwarzschild radius. have Allowing the objects to without havelar angu- momentum being (spin) pushes the a limit down by black a factor hole is its as described in this subsection, asnamics, as well described as in the the orbital next dy- subsection. The compactness ratio can also be defined in relation to tem becomes more compact as the massthat ratio (for increases. We note mass ratio, the system becomes morecentricity compact until with growing ec- ( tion between the two objectsradii) to as the a sum function of of theirthe mass Schwarzschild very ratio high and (arbitrary) eccentricity value from of ( 150 Hz r r Original Paper 8o 7 1600209 17) of (8 we why is This 7. grav- Eq. the by used by given driven emission not wave thus itational are revolutions final frequency the and in separation orbital in 911 changes pp. The [18]). (see of inwards “plunge” and trajectories non-circular interior be Allowed must below. typi- given whose is (ISCO), location orbit cal circular past is stable orbits objects such innermost no the the are there of other) one the than when larger least much (at merger objects the compact to between close of inspiral relativity, general the in drives However them. energy waves and gravitational merger, to to down way lost the circular all stable exist dynamics may Newtonian orbits In or- not: last does the for it fact bits, In [31]. mass regime chirp non-Newtonian the the might describes in correctly one 7 compact, Eq. whether were ask bodies still hav- merger, the before that cycles accepted final ing the analyzing are we As on constraints – measured? well mass chirp the Is 4.5 sue htteobti o-opc,te u analysis our then non-compact, is orbit the that assumes compactness to down valid is approximation Newtonian the o-elgbeol nteobt eycoet black conclu- a our refute to not sions. does close argument very this again are orbits so coordinates hole, the these in using only in How- non-negligible errors freedom. the gauge too, to here due ever sub- arbitrariness also some are to separations ject The general inaccurate. entirely be not are might these and of both as caption), 4 its Fig. and (see radii object compact corresponding the co- to tion the using to and/or [15] ordinate formula quadrupole the of significant are orbits. compact thus for and only [28–30]), respectively 2PN, (1.5PN parameter and PN the of power a with suppressed also are these However, interactions. adding spin-spin light, and effectively spin-orbit of dynamics, speed Newtonian the the approach modifying also may velocity tational compact. is orbit con- the the that to clusion leads and an relativity as general justified of is approximation mechanics Newtonian using data the of eitl eati em ftecmatesratio, compactness the of terms in so recast radius, mediately Schwarzschild the includes ter f 2 GW R h aeraoigmyas eapidt h use the to applied be also may reasoning same The ro- their spinning, rapidly is bodies the of either If h xrsinfrtedmninesP parame- PN dimensionless the for expression The h niiulmasses individual the fin R − f 1 GW . sNwoindnmc od when holds dynamics Newtonian As . fodro e.Agigb otaito,i one if contradiction, by Arguing few. a of order of R max o h oprsno h elra separa- Keplerian the of comparison the for ttepa,rte hntefia frequency final the than rather peak, the at www.ann-phys.org x C a eim- be can 06TeAuthors. The 2016 x ssmall, is x ∼ nae e Physik der Annalen r q h nems tbecrua ri wihi at is at (which orbit frequency circular wave stable gravitational innermost quadrupole the the 0) c ls nodrfrtesalrmass smaller the for order in cles, given because chosen value evtv eto trms pe on t4 at bound upper mass star neutron servative 6 ( hole black Schwarzschild a around example, For unyfrapug into plunge a for quency threshold, erbakhl tcicdswt h nems tbecir- at stable orbit cular innermost extremal the an with spin coincides For it the decreases. hole black radius as Kerr ring hole, light black the Kerr creases spinning a for while ω [32] by radius given at is orbit coordinates) Boyer-Lindquist equatorial (in circular, a of infinity at sured 8 out going at is ring ring light light The the . at [33–37] barrier encounter potential ring effective light an the within originating waves as and bevdaoe3M 3 been above have observed stars neutron No stage. late the at proximation n h uduoegaiainlfeunyis frequency gravitational quadrupole the and hsrdu s3 is radius This cl ihteivreo t as n loivleisdi- its involve also and spin mass, mensionless its of inverse the object with an scale around waveforms) (hence orbits test-particle i rudtelre as( or- mass geodesic larger a the around follows bit approximately mass where (EMRI), smaller inspiral the ratio the mass of extremal treatment an a as suggests system ratio mass high a Such have? hl o netea erbakhl ( hole black Kerr extremal an for while ohn hscli xetdt ri atrta light, than faster orbit to expected as is (LR), physical ring light nothing the is from frequency frequency expected the highest approximately the plunge, final the from on ia rqec tIC ( ISCO at frequency bital .P bo ta. h ai hsc ftebnr lc oemre GW150914 merger hole black binary the of physics basic The al.: et Abbot P. B. LR 3 GM orb ≥ / et aeas ensonb Rt easn [11–13]. absent be to NR by shown been also have fects ef- GR nonlinear to due up-conversions frequency Hypothesized 2 esalnwcntanteidvda assbased masses individual the constrain now shall We = f 0.I uhahigh a such Is 100. π GW = / GM 2 c 2 r GM fin c shneeulto equal hence is ) 3 2 / 2 o hc ed o edteNwoinap- Newtonian the need not do we which for , = m + 32(M ulse yWlyVHVra mH&C.Ka Weinheim KGaA Co. & GmbH Verlag Wiley-VCH by published GM 1 χ 1 √ utb tlat46M 476 least at be must + GM GM / √ c cos χ GM 2 / h ria frequency orbital The . h aia rvttoa aefre- wave gravitational maximal The . / M esalrl na vnmr con- more even an on rely shall we ; c 2 / H.Fragaiainlwave gravitational a For kHz. ) 3 2 q c o cwrshl lc hole, black Schwarzschild a for m cos r osbewt h aata we that data the with possible 3 1 = m = ste 7Hz. 67 then is M − 1 GM 1 GM ( ∼ c − rmteeryvsbecy- visible early the from f 3 GW χ / M c ) 2 .Tefeunisof frequencies The ). m = )is χ 2 4 + . ob eo this below be to . hc implies which , 4(M ω χ orb GM c ω = 2 orb r = )teor- the 1) / . M c 6M 76 smea- as 3 3 / )kHz, / 2 f 2 χ GW GM (16) (17) − χ r in- 1 ,a . = = = r 8 , Original Paper ∼ 6 (19) (20) (22) (21) 9 5 . , with 0 6 , classical 2 ω × , 4 2 L r − d 2 . 6 for the final max M 5 × (9 of 17) 1600209 G h c G 4 L c / . d 3 ∼ c L GW ∼ ω . (as defined in App. A) . for an equal-mass sys- 1, and so does not sub- ω , . 300 Mpc. This distance 2 W 2 5 0 M 1. Also, analysis of a small . Taken together with the ∼ orb ∼ c 52 h ≤ 21 E 3 ∼ L 5 L . − z − max d 10 be called the “Dyson luminosity” in has been proposed as the upper d q 0 c h 10 10 G G × ,and ∼ GW 4 / 2 6 × 5 / ω . r c at luminosity distance 5 c 3 / 2 ω c acquires a factor 0 . h 0 = L 5 G max c ,whichgives GM ∼ www.ann-phys.org 4 G 2 and Hz / ∼ GW 5 ∼ f G μ M c r / max 2 Planck = 5 32 L ˙ h L . While the numerical value may change by a ISCO 3 and r 2 L − 2 ≡ d G r c Planck 3 10 / 1 6 4 45Gpc c L c × Using the orbital energy We can obtain a more accurate distance estimate Using Eq. 5 we relate the luminosity of gravitational ∼ ∼ peak honor of the physicist Freeman Dyson and because itquantity is that a does not contain the Planck constant limit on the luminosity of any physical system [38–40]. Gibbons [41] has suggested that The “Planck luminosity” ∼ ∼ ∼ Planck 2 L . L L M L L stantially affect any ofdistance-luminosity calculation the based conclusions. only on For thedata a strain (reaching a different similar estimate), see [42]. based on the luminosity, because theluminosity gravitational wave from anpeak value equal-mass which is binary independent of the inspiral mass.seen This from can has naive be dimensional a analysis offormula, the quadrupole which gives a luminosity we may also estimate theitational total waves energy radiated during as grav- the system’s evolution from a 9 tem, and is close to that for object falling into a Schwarzschild black hole suggests 0 However, a closershould look be (Eq. A4) shows the prefactor Thus we have correct exponents, corresponds to a redshift of waves to their strain factor of a few with theder specific of spins, magnitude we as can universal treat for its similar-mass or- binaries. which for GW150914 gives and we estimate themeasured strain in distance time over from the cycle at theas peak amplitude, change of the d ω tight orbit. Together this gives the Planck luminosity, . 21 − is the 10 z ∼ .Asshown max L d h / 1 falls off with in- ∝ h h as reached unity, because L h d published by Wiley-VCH Verlag GmbH & Co. KGaA Weinheim 6 Gpc (18) ∼ , No. 1–2 (2017) , beyond the maximum observed mass of Annalen der Physik 529 200 km 3M × ) relative to the detector frame, where > z 21 1, so the detector- and source-frame masses differ + 10 . 200 km the non-linear nature of gravity would be- the luminosity 0 Because we see gravitational wave emission from or- The gravitational wave amplitude < ∼ 76M 2016 The Authors. ≤ L . C R z by less than of order 10%. 4 bital motion at frequencies muchimal higher value, than with this or max- withoutout. spin, Hence even such the lighter a of system the masses is must ruled be at least on the distance to the source. creasing luminosity distance tance from us, and thetional waves. total energy radiated in gravita- Basic physics arguments alsopeak gravitational provide wave luminosity estimates of of the system, the its dis- 5 Luminosity and distance the gravitational wave from thestrain and merger flux to at the the detector. observed The redshift is found to be differ from their values at the source?we In the estimate next section, the distanceredshift, to by the relating source the and hence amplitude the and luminosity of by (1 redshift. Direct inspectionmass of values the from detector the red-shifted data waves. yields How do these gravitational wave phasing correspondsthe to masses a as measured scaling onof of Earth; Eq. dimensional 7 analysis shows that the source frame masses are smaller observed onemitted. Earth The same compared effect accounts tophotons for from the distant their redshifting objects. of The values impact of when this on the Gravitational waves are stretchedthe Universe by as the they expansionwavelength travel of across and it. decreases This the increases frequency the of the waves 4.6 Possible redshift of the masses – a constraint from Ann. Phys. (Berlin) magnitude upper bound d near the Schwarzschild radius of the combined system come apparent. In this way we obtain a crude order-of- the measured strain wouldtimes have larger. This peaked could be at continued, alationship but the value would scaling break ten re- before Had our detector been ten times closer to the source, neutron stars. in Fig. 1, the measured strain peaks at Original Paper q oeshv enue o siaigteprmtr of parameters the estimating and for used we analyses been such and have how models information, explore to more reader even the encourage us give can methods, techniques. advanced application more without of data strain the in seen be there- cannot and fore but smaller is amplitude detection, 4.5, whose wave 45], [6, Sec. gravitational GW151226 another out in been There rule already done GW150914. to has safely similar as systems to for useful constituent low be should star too are neutron masses a the if instance the from constrained further be 3]. [2, can data value its thus 44], 1 f1)1600209 17) of (10 on depend not does 7 Eq. by described evolution quency ratio mass the about conclusions ited [43]). see approach phe- related nomenological a (for properties basic hole. and the black distance system’s about single information us a give to also can likely arguments most Simple down, settle to seen holes is system The black merging. before inspiraling closely of very the approached pair that of produced a data that was system strain wave gravitational the ba- observed show these the These applying to by GW150914. arguments obtained be physics can sic insight of lot A Conclusions 6 from output sun. power our the than greater magnitude of orders and (where separation initial large very i.1,btfrteccea eklmnst,ispower its luminosity, peak at cycle the in for shown but second 1), a of Fig. fraction the over entirely (almost mass release its GW150914 did of only ∼ 1% Not than radiation. less and convert light into to expected is sun lifetime, ten-billion-year our its During remarkable. is event with compare [1–3]. ringdown); in calculations and exact merger the the a in for energy emitted some (as estimate is energy emitted an total the considered on bound be lower should quantity This separation L E .Themassratio peak GW 0 ie smc nryi rvttoa waves gravitational in energy much as times 300 uhtcnqe,cmiigaayi n numerical and analytic combining techniques, Such for signal, every for work not will arguments These ihteebscagmnsw aeol rw lim- drawn only have we arguments basic these With ent htteaon feeg mte nthis in emitted energy of amount the that note We r = ntefr fgaiainlwvswsaot22 about was waves gravitational of form the in ∼ E R orb i = r − o W594 using GW150914, For . 5 m(q 9), (Eq. km 350 E orb f q osapa nteP orcin [22, corrections PN the in appear does = 0 − www.ann-phys.org − GM 2 R μ E ∼ orb i q m 3M eas h fre- the because , 1 → ∼ )dw oa to down 0) m c C 2 2 06TeAuthors. The 2016 . ∼ 5M 35 (23) nae e Physik der Annalen eew uln h aclto fteeeg binary a energy the of calculation the outline we Here tuto n prto fteVrodtco n h rainand creation the and detector Virgo the of con- operation the and for struction Research, Scientific for Organisation Netherlands and by the supported (CNRS) Matter on Scientifique Research Fundamental Recherche for Foundation the la the (INFN), de Nucleare National Fisica Centre di French Nazionale Istituto Italian acknowl- the gratefully edge authors The by Council. provided Research was Australian LIGO the Advanced for GEO600 support the of Additional operation detector. and construction and construction LIGO Advanced the the of of and support (MPS), for Niedersachsen/Germany Max-Planck-Society of the State Kingdom, United the (STFC) Council of Facilities Technology and Science the Advanced as well and as Laboratory LIGO LIGO the of operation and the for construction (NSF) Foundation Science National States United the of port Acknowledgements. wave gravitational of era observations. the of beginning the at field, the this on based studies astrophysical [5]. event for and dynamic [4] relativistic, highly valid- regime the the in constraining relativity general and of testing ity for 3], [2, system the fteNF TC P,IF,CR n h tt fNiedersach- of State the resources. computational and of CNRS provision for INFN, sen/Germany MPS, STFC, NSF, the of support the the acknowledge gratefully and authors Taiwan The (MOST), Foundation. Kavli Technology Corpora- and Research Science of the Ministry Trust, tion, Leverhulme the Research, Basic S for de Estado do Fundac Pesquisa Innovation, and Technology, Ministry Science, Brazilian Canada, the of Research, Council Advanced Research for Institute Engineering Canadian and Science Innovation, Natural and Development the Economic of Ministry Ontario the of Province through the and Research Canada National Industry the Korea, of (LIO), Foundation Origins of Institute Lyon Fund the Research (OTKA), Scientific Hungarian the Uni- Alliance, Scottish Physics the Council, versities Funding Scottish the Society, Commission, Royal European the the Poland, of Centre Science National the Conselleria and Competitivitat d’Educaci i d’Economia Conselleria the dad, Econom de Ministerio Devel- Spanish Resource Engineering the Human India, & opment, of Ministry Science India, India, (SERB), Technology, Board Research and India, Depart- Science of by of as Research ment Industrial well and as Scientific of agencies ac- Council these gratefully the from also support authors research The consortium. knowledge EGO the of support ytmeisi rvttoa ae n h mte en- system. the emitted on the effect and ergy’s waves gravitational in emits system aito rmabnr system binary a from radiation gravitational of Calculation A: Appendix .P bo ta. h ai hsc ftebnr lc oemre GW150914 merger hole black binary the of physics basic The al.: et Abbot P. B. ehp htti ae ilsrea nivtto to invitation an as serve will paper this that hope We ,CluaiUiestt fteGvr elsIlsBalears, Illes les de Govern the of Universitats i Cultura ´ o, ulse yWlyVHVra mH&C.Ka Weinheim KGaA Co. & GmbH Verlag Wiley-VCH by published h uhr rtflyakoldetesup- the acknowledge gratefully authors The ˜ oPuo(AEP,RsinFoundation Russian (FAPESP), Paulo ao ¸ ´ ˜ od Amparo de ao ayCompetitivi- y ıa ` a Original Paper ’s ∝ 4. . r h 3 (A5) (B1) , and its R 3 2 2 Gm c , kg m . 5 5 / /ω = 2 1 2 M R (11 of 17) 1600209 GM G M 3 = . m driven by emission μ 2 11 3 15 r c r 35 M ω / = 3 L not 3 ∝ M 5 96 4 ω . R 3 = ) grows by at least a factor of 5. t we can substitute for all the 2 ( 15 ω M ,andso 3 10 ˙ ω μ ω/ω × 2 5 r www.ann-phys.org 2 3 3 G Mr = 11 15 . α c ω 3 =− ω , which are constrained to fit into a radius = 3 R r L ) should be decreasing. For a system not driven L 3) t ∝ m ( 5 96 r 2 π/ w 35 M (4 , in order to describe each orbit as approximately Ke- system 2 Thus we see that for a system Now using Kepler’s third law We can see that Eq. A5 describes the evolution of The quadrupole formula (Eq. 4) then indicates the = ∼ such that the compactness ratio obeys ≥ shrinks, both its gravitational wave frequency and am- orb 3 ˙ R We are considering astrophysical objects with mass scale E Mr of gravitational waves, as the characteristic system size plitude grow, but remain proportional to each other.is This inconsistent with thewhich show data the amplitude of only grows by GW150914 a factor2 (Figs. of while about the 1, frequency 2), under different mechanisms. Anmeans increase the in system frequency it rotates gains faster angular momentum, and thelength faster, system’s characteristic so unless Appendix B: Possibilities for massive, compact objects A.1 Gravitational radiation from a different rotating A rising gravitational wavea rise amplitude in can frequency accompany in other rotating systems, evolving having defined the chirp mass m This produces a scale for their Newtonian density, ρ plerian. derivative ˙ and obtain ω the system as an inspiral:(“chirps”), the while orbital by frequency Kepler’s goes Lawshrinks. up the orbital separation gravitational wave strainsecond amplitude time should derivative of follow the the quadrupole moment, by the loss oftational energy waves, and rapidly losing angular angular momentum momentumdifficult, is to thus also the gravi- system should conserve its angularmentum mo- r , = = μ r ) 2 t yx (A3) (A4) (A2) (A1) )de- GM ( I -plane x of the ij ( , = ij xy Q ρ ⎞ ⎟ ⎟ ⎠ =− Q xy 2 A I r orb 0 0 1 3 ), t E − and frequency ω 2 r A 2 x + 1 3 orbiting in the ) whose origin is the cos(2 A 1 z 2 y r , − − A m y x 1 3 A 2 = , y . We assume that the en- r x ij 3 2 ( = and δ published by Wiley-VCH Verlag GmbH & Co. KGaA Weinheim GW 2 1 yy = r -plane. In the simple case of E I A 2 m ) t 3 1 y d , 3 d xy 1 3 x 1 3 the radial distance from the ori- . Combining we find A − , 2 3 y . j 2 r 00 + 6 − A x x =− ) x ω i A 2 , t 1) is the Kronecker-delta and 4 ˙ x r 1 x , =− r ω in the x 2 3 μ 2 2 1 , ( r ) } , No. 1–2 (2017) zz , μ 2 ij I ⎜ ⎜ ⎝ ⎛ x Annalen der Physik 2 GM I = 5 ( , A 529 G c 2 A ρ x 1 cos(2 r = m x } A 2 5 2 3 32 = , ∈{ diag(1 1 A two-body system, d m orb xx A = ∈{ E )and = , and the gravitational wave luminosity from Eq. , a little trigonometry gives for each object (see I A t t = ij d π d ω ij I ω ) 2 GW δ t = = 2 First we calculate the quadrupole moment ( r E = A ij ij 2016 The Authors. t μ d f C d 2 1 Q Q gin. center-of-mass, with 5is where Figure A1 Ann. Phys. (Berlin) This energy loss drains the orbital energy thus ergy radiated away over each orbit is small compared to bodies a circular orbit at separation where the second equality holds for a system of two Fig. A1) around their C.O.M. system’s mass distribution. We usenate system a Cartesian coordi- sin(2 notes the mass density. Then Original Paper e fQM seueae yvrosdsrt indices, determined are discrete times damping various and frequencies by their and enumerated is QNMs The of (QNMs). set quasi-normal-modes characteristic thus have should wave gravitational emitted the and linearize, should perturbations remaining the stage, ringdown this etedw oafia tt faKr lc oe uniquely hole, black mass Kerr its by a [50] of defined state final a to and hair” down its settle “lose to proceeds It distorted. hole very black is a horizon merger, a in formed being after Immediately distinguish to off hard clearly. level becomes amplitude to signal the seems as frequency just The dropping amplitude. while in Hz, sharply 250 about in reaching rise until to continuing frequency cycles, additional two to signal one the makes reached, is amplitude wave gravitational peak signal interpretation. the this also to this support of that lends argue properties and frequencies, subsequently the higher at they discuss waveform now that We closely holes, merged. black so two approached been had source which the that shows Hz 150 els es.Wieti est safco of factor a would is which density this star, While main-sequence dense. any less be as well pres- as degeneracy electron sure, by supported objects out rule 1 f1)1600209 17) of (12 frequencies wave gravitational for sig- the waveform of nal properties observable directly the that ments, final object? the and ringdown the can about we conclude what phase: inspiral Post C: Appendix holes. black are compactness and mass such with ob- likeliest jects The continuum. stellar un- the and in niche unexplored observed heretofore and narrow occupy extreme, to an need would they these bodies, of any material if were that objects show do data the that conclude can we as low as ratio compactness a theoretically narrower although even Thus, an corner. into state of equation thus the density, constraining neutron-star to these closer push even to bodies would order massive This in frequency. compact orbital more final the even reach be to bodies re- the undoubtedly quired have would change, distortions, frequency tidal the including of analysis careful more A [48]). in a is This 10 object. of uniform a factor for attained is equality where mmnurnsa asb nodro antd,as magnitude, of is order limit an star neutron by the mass max- star the exceed neutron bodies imum these stars, neutron than dense ehv rud sn ai hsc n cln argu- scaling and physics basic using argued, have We sti ossetwt egrrmatbakhole? black remnant merger a with consistent this Is the after that show 2 and 1 Figures in data The R 6 = oedneta ht wrs ow can we so dwarfs, white than dense more 4 / spritdfruiomojcs[49], objects uniform for permitted is 3 ∼ M www.ann-phys.org 3M n pnparameter spin and 32M (3.2 n[6 7,29M 2.9 47], [46, in C 06TeAuthors. The 2016 ∼ χ aein Late . 10 f GW 2 less < nae e Physik der Annalen u oteodro-nt factor order-of-unity mass the the to by or- determined (up thus between is 2 frequency gravi- wave of final tational factor The frequencies. a wave gravitational with and 17), bital (16, Eqs. from ately edn siaeo h asadsi [51]. spin and inde- mass an the give of estimate could a pendent perturbations, with hole conjunction black in for time, model analyzing damping found, and mode frequency a its such Were frequency? waveform sinusoid fixed exponentially-damped observed of an the out of of dying evidence end and contain the peak, Does the amplitude. after cycles in two about Hz) 250 around (at frequency see in stabilizing clearly wave do gravitational We the formed. evidence was strong remnant final several be single would of a that frequency ringdown fixed a a finding with so cycles and – mode damped) by h ih ig adtn uhlwrfeunis.For frequencies). lower parameter much the mandating ring, light than farther the much radius orbiting larger bulges of distortion have (objects and would objects mass the total of compactness the our the constrains how show frequency to attained this used high already fact in have We spin). iesols ope number complex dimensionless a e found The then from are time. times decay damping (inverse) and the amplitude ringdown is part imaginary the and hr h elpr of part real the where obe to cltbltoso h Ns[2 hwthat show [52] QNMs the of tabulations ical on However, values [52]. limiting a eigenfrequencies obtain complex to of conditions set discrete boundary the equations, enforces field one the then cavity: solve and to resonant variables a of separation of uses modes normal the analyzing o uduoa oe( mode frequency quadrupolar expected a the for refer- 17)); and (16, [52] Eqs. and (See therein, horizon ences hole black the re- ring outside as light a of orbit on thought traveling space-time be of can distortion a hole to lated black Kerr a of ringing The analysis Mode C.1 c G f .P bo ta. h ai hsc ftebnr lc oemre GW150914 merger hole black binary the of physics basic The al.: et Abbot P. B. i GW 3 ω GW M M h xc ausof values exact The | ringdown t ω = and f GW GW e i | = ringdown GM c χ 3 x ahsc e ol aealaig(least- leading a have would set such Each . τ x t ulse yWlyVHVra mH&C.Ka Weinheim KGaA Co. & GmbH Verlag Wiley-VCH by published damp e + − y GM c i eemnn h apn ie numer- time, damping the determining 3 y y = = . t x = c , 2 3 x π x x e / 2 y ∈ π x ω i 2 ∼ ( f ∼ π and GW GW = (C3) 1 GM | 0 ringdown m . 3 steaglrfrequency angular the is y , = ] r eie immedi- derived are 1], x a efuda when as found be can and t )wl hsb ie as given be thus will 2) hc moisthe embodies which , e − t τ /τ damp damp , = GM / c 3 (C2) (C1) y . Original Paper ,3 17 , 204 , 908– 7 , 1075– , L51–L53 253 , 435–439 692 (2) (2014). 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(16, 17) estimate that . , No. 1–2 (2017) . 0 65 M Annalen der Physik 0 260Hz 529 ∼ ∼ ≈ ] GW150914, gravitational waves, black holes. χ GW 4ms ω , giving a ringdown frequency Such a final mass is consistent with the merger of i = (2016). (2016). arXiv:1606.01210 [gr-qc]. (2016). arXiv:1606.04856 [gr-qc]. M ringdown 3 | G c The exact value can be found using Table II in [52], While it is beyond the scope of this paper to calcu- 0808 2016 The Authors. GW . [4] B. P. Abbott et[5] al., Phys. B. Rev. P.Abbott Lett. et al., Astrophys. J. [2] B. P. Abbott et[3]B.P.Abbottetal.,Phys.Rev.X al., Phys. Rev. Lett. [1] B. P. Abbott et al., Phys. Rev. Lett. 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IO aionaIsiueo ehooy aaea A915 USA 91125, CA Pasadena, Technology, of Institute California LIGO, mrcnUiest,Wsigo,DC,206 USA 20016, D.C., Washington, University, American IOLvnso bevtr,Lvnso,L 05,USA 70754, LA Livingston, Observatory, Livingston LIGO -02 aoi Italy Napoli, S.Angelo, I-80126 Monte di Universitario Complesso Napoli, di Sezione INFN, oiin tt nvriy ao og,L 00,USA 70803, LA Rouge, Baton University, State Louisiana France Universit (LAPP), Particules des Physique de d’Annecy-le-Vieux Laboratoire nvriyo lrd,GievleF,361 USA 32611, FL, Gainesville Florida, of University NN ein iNpl,I810 aoiItaly Napoli I-80100, Napoli, di Sezione and INFN, Italy Benevento, I-82100 Benevento, at Sannio of University Universit eesd ac 2015. March Deceased, ihf cec ak 08X,Asedm h Netherlands The Amsterdam, XG, 1098 Park, Science Nikhef, eesd a 2015. May Deceased, pyi,D146PtdmGl,Germany Potsdam-Golm, D-14476 sphysik, f Max-Planck-Institut Albert-Einstein-Institut, S Campos, S 12227-010, Espaciais, Pesquisas de Nacional Instituto aetlRsac,Bnaoe501,India 560012, Bangalore Research, Fun- damental of Institute Tata Sciences, Theoretical for Centre International NN ein iRm o egt,I013Rm Italy Roma I-00133 Vergata, Tor Roma di Sezione INFN, NN ein iPs,I517Ps,Italy Pisa, I-56127 Pisa, di Sezione INFN, USA 02139, MA Cambridge, Technology, of Institute Massachusetts LIGO, nvriyo icni-iwue,Mlake I521USA 53201 WI Milwaukee, Wisconsin-Milwaukee, of University 107 India 411007, Pune Astrophysics, and Astronomy for Centre Inter-University ebi Universit Leibniz NN rnSsoSineIsiue -70 ’qia Italy L’Aquila, I-67100 Institute, Science Sasso Gran INFN, pyi,D317 anvrGermany Hannover D-30167, sphysik, f Max-Planck-Institut Albert-Einstein-Institut, h nvriyo issip,Uiest,M 87,USA 38677, MS University, Mississippi, of University The Universit France Universit CNRS/IN2P3, Paris-Sud, Univ. LAL, oy00,Australia 0200, tory Terri- Capital Australian Canberra, University, National Australian dom King- United 1BJ, SO17 Southampton Southampton, of University hna ahmtclIsiue hna 013 India 603103, Chennai Institute, Mathematical Chennai Universit NN ein iRm,I015Rm,Italy Roma, I-00185 Roma, di Sezione INFN, Universit aionaSaeUiest ulro,Fletn A981 USA 92831, CA Fullerton, Fullerton, University State California eesd ac 2016. March Deceased, 12 .Yamamoto H. , 122 ´ aoeMn ln,CR/NP,F791Annecy-le-Vieux, F-74941 CNRS/IN2P3, Blanc, Mont Savoie e ` iSlro icao -48 aen,Italy Salerno, I-84084 Fisciano, Salerno, di a ` iRm o egt,I013Rm,Italy Roma, I-00133 Vergata, Tor Roma di a ¨ tHmug -26 abr,Germany Hamburg, D-22761 Hamburg, at ` iPs,I517Ps,Italy Pisa, I-56127 Pisa, di a .Zhang Y. , 1,12 ˜ oPuo Brazil Paulo, ao 43 ,S.E.Zuraw .Zanolin M. , ¨ tHnoe,D317Hnoe,Germany Hannover, D-30167 Hannover, at 101 1 .Zhao C. , ,C.C.Yancey 118 97 n .Zweizig J. and , .P Zendri J.-P. , www.ann-phys.org 51 .Zhou M. , 63 ,H.Yu 43 ,M.Zevin 81 1 .Zhou Z. , 12 ´ ai-aly Orsay, Paris-Saclay, e ,M.Yvert rGravitation- ur ¨ rGravitation- ur ¨ 81 ,L.Zhang 81 ,X.J.Zhu 8 .Zadro A. , C ˜ oJos ao 06TeAuthors. The 2016 1 ´ dos e , 51 , ˙ zny 112 , nae e Physik der Annalen 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 66 65 64 63 62 61 60 59 58 57 56 55 54 53 52 51 50 49 48 .P bo ta. h ai hsc ftebnr lc oemre GW150914 merger hole black binary the of physics basic The al.: et Abbot P. B. Australia 6009, Australia Western Crawley, Australia, Western of University Poland Warsaw, 00-478 University, Warsaw Observatory Astronomical tnodUiest,Safr,C 40,USA 94305, CA Stanford, University, Stanford Universit UA nvriyo lso,GagwG28Q ntdKingdom United 8QQ, G12 Glasgow Glasgow, of University SUPA, Italy Pisa, Cascina, I-56021 (EGO), Observatory Gravitational European 42,Nc ee ,France 4, cedex Nice 34229, Universit Artemis, NN ein iFrne -01 et irnio iez,Italy Firenze, Fiorentino, Sesto I-50019 Firenze, di Sezione INFN, ahntnSaeUiest,Plmn A914 USA 99164, WA Pullman, University, State Washington -50 ai ee 3 France 13, Cedex Paris F-75205 Cit Paris Sorbonne Paris, de Observatoire CEA/Irfu, CNRS/IN2P3, Universit Cosmologie, et AstroParticule APC, ..Bx91,60 L imgn h Netherlands The Nijmegen, GL, 6500 9010, Box P.O. Nijmegen, University Radboud Astrophysics/IMAPP, of Department IOHnodOsraoy ihad A932 USA 99352, WA Richland, Observatory, Hanford LIGO yaueUiest,Srcs,N 34,USA 13244, NY Syracuse, University, Syracuse nvriyo ayad olg ak D272 USA 20742, MD Park, College Maryland, of University nvriyo rgn uee R943 USA 97403, OR Eugene, Oregon, of University Hungary Mikl Thege Konkoly Budapest, H-1121, RMKI, RCP, Wigner ilubne France Villeurbanne, Mat des Laboratoire Italy Universit France Coll University, Research ENS-PSL Universit UPMC-Sorbonne Brossel, Kastler Laboratoire UUiest mtra,18 V mtra,TeNetherlands The Amsterdam, HV, 1081 Amsterdam, University VU NN ein iGnv,I116Gnv,Italy Genova, I-16146 Genova, di Sezione INFN, nvriyo imnhm imnhmB52T ntdKingdom United 2TT, B15 Birmingham Birmingham, of University Italy Padova, I-35131 Padova, di Sezione INFN, Italy Perugia, I-06123 Perugia, di Sezione INFN, Universit UA nvriyo h eto ctad ase A B,United 2BE, PA1 Kingdom Paisley Scotland, of West the of University SUPA, -54 ens France Rennes, F-35042 Universit CNRS, Rennes, de Physique de Institut aoa Italy Padova, Universit RA,Idr,M 503 India 452013, MP Indore, RRCAT, 191 Russia 119991, Moscow University, State Moscow Lomonosov Physics, of Faculty Universit nttt fTcnlg,Alna A332 USA 30332, GA Atlanta, Technology, of Georgia Institute Physics, of School and Astrophysics Relativistic for Center altnClee otfed N507 USA MN 55057, Northfield, College, Carleton oubaUiest,NwYr,N 02,USA 10027, NY York, New University, Columbia AKPN 076Wra,Poland Warsaw, 00-716 CAMK-PAN, ` iPrga -62 eui,Italy Perugia, I-06123 Perugia, di a ` el td iUbn CroB” -12 Urbino, I-61029 Bo”, “Carlo Urbino di Studi degli a ´ lueBradLo ,F662Vleran,France Villeurbanne, F-69622 1, Lyon Bernard Claude e ` ` iPdv,Dpriet iFsc srnma I-35131 Astronomia, e Fisica di Dipartimento Padova, di a el td iGnv,I116Gnv,Italy Genova, I-16146 Genova, di Studi degli a ulse yWlyVHVra mH&C.Ka Weinheim KGaA Co. & GmbH Verlag Wiley-VCH by published ´ eC ´ raxAvanc eriaux t ’zr NS bevtieC Observatoire CNRS, d’Azur, ˆ ote ´ s(M) NSI23 F-69622 CNRS/IN2P3, (LMA), es ` g eFac,F705Paris, F-75005 France, de ege ´ ai Diderot, Paris e ´ eRne 1, Rennes de e t ’zr CS d’Azur, ˆ ote ´ s CNRS, es, ´ os t29-33, ut ´ ´ e, Original Paper ora,´ 65- (17 of 17) 1600209 ao Paulo, SP 01140- ˜ lystok, Poland orica,´ University Estadual Paulista/ICTP South www.ann-phys.org lystok, 15-424, Bia ora,´ Poland ısicaTe Swierk-Otwock, Poland ´ ´ a di Camerino, Dipartimento di Fisica, I-62032 Camerino, e de Lyon, F-69361 Lyon, France ` ´ a di Siena, I-53100 Siena, Italy ` University of Bia Pusan National University, Busan 609-735, Korea IISER-TVM, CET Campus, Trivandrum, Kerala 695016, India Institute of Applied Physics, Nizhny Novgorod, 603950, Russia Instituto de F American Institute for Fundamental Research, S 070, Brazil Universit Italy SUPA, University of Strathclyde, Glasgow G1 1XQ, United Kingdom Universit Andrews University, Berrien Springs, MI 49104, USA Canadian Institute for Theoretical Astrophysics, University of Toronto, Toronto, Ontario M5S 3H8, Canada National Institute for Mathematical Sciences, Daejeon 305-390, Korea IISER-Kolkata, Mohanpur, West Bengal 741252, India Whitman College, 345 Boyer Avenue, Walla Walla, WA 99362, USA College of William and Mary, Williamsburg, VA 23187, USA Abilene Christian University, Abilene, TX 79699, USA Trinity University, San Antonio, TX 78212, USA Kenyon College, Gambier, OH 43022, USA University of Cambridge, Cambridge, CB2 1TN, United Kingdom Janusz Gil Institute of Astronomy, University of Zielona G Rutherford Appleton Laboratory, HSIC, Chilton, Didcot, Oxon OX11 0QX, United Kingdom Montana State University, Bozeman, MT 59717, USA Universit University of Washington, Seattle, WA 98195, USA King’s College London, University of London, London WC2RUnited 2LS, Kingdom 265 Zielona, G Hanyang University, Seoul 133-791, Korea Monash University, Victoria 3800, Australia ESPCI, CNRS, F-75005 Paris, France The Chinese University of Hong Kong, Shatin, NT,China Hong Kong SAR, University of Massachusetts-Amherst, Amherst, MA 01003, USA NCBJ, 05-400 Southern University and A&M College, Baton Rouge, LA 70813, USA Seoul National University, Seoul 151-742, Korea University of Alabama in Huntsville, Huntsville, AL 35899, USA Hobart and William Smith Colleges, Geneva, NY 14456, USA IM-PAN, 00-956 Warsaw, Poland University of Adelaide, Adelaide, South Australia 5005, Australia 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 published by Wiley-VCH Verlag GmbH & Co. KGaA Weinheim er 9, Szeged 6720, Hungary ´ om´ t , No. 1–2 (2017) Annalen der Physik 529 os¨ University, “Lendulet” Astrophysics Research Group, a di Trento, Dipartimento di Fisica, I-38123 Povo, Trento, ` a di Roma “La Sapienza”, I-00185 Roma, Italy a di Napoli “Federico II”, Complesso Universitario di Monte ` ` otv ¨ University of Michigan, Ann Arbor, MI 48109, USA NCSA, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA lorca, Spain Universitat de les Illes Balears, IAC3—IEEC, E-07122 Palma de Mal- 2016 The Authors. Rochester Institute of Technology, Rochester, NY 14623, USA Cardiff University,Cardiff CF24 United 3AA, Kingdom 3FD, United Kingdom School of Mathematics, University of Edinburgh, Edinburgh EH9 Sonoma State University, Rohnert Park, CA 94928, USA Institute for Plasma Research, Bhat, Gandhinagar 382428, India INAF, Osservatorio Astronomico di Capodimonte, I-80131 Napoli, Italy University of Szeged, D RESCEU, University of Tokyo, Tokyo, 113-0033, Japan Tata Institute of Fundamental Research, Mumbai 400005, India INFN, Trento Institute for Fundamental Physics and Applications,38123 I- Povo, Trento, Italy Tsinghua University, Beijing 100084, China West Virginia University, Morgantown, WV 26506, USA NASA/Goddard Space Flight Center, Greenbelt, MD 20771, USA Budapest 1117, Hungary MTA E Universit Italy University of Minnesota, Minneapolis, MN 55455, USA Indian Institute of Technology, Gandhinagar Ahmedabad, Gujarat 382424, India Embry-Riddle Aeronautical University, Prescott, AZ 86301, USA University of Brussels, Brussels 1050, Belgium National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan The University of Sheffield, Sheffield S10 2TN, Kingdom United Australia Charles Sturt University, Wagga Wagga, New South Wales 2678, Caltech CaRT, Pasadena, CA 91125, USA Universit The University of Melbourne, Parkville, Victoria 3010, Australia S.Angelo, I-80126 Napoli, Italy Universit USA The University of Texas Rio Grande Valley, Brownsville, TX 78520, The Pennsylvania State University, University Park, PA 16802, USA University of Chicago, Chicago, IL 60637, USA Korea Institute of Science and Technology Information, Daejeon 305-806, Korea National Tsing Hua University, Hsinchu City 30013, Taiwan,of Republic China Montclair State University, Montclair, NJ 07043, USA (CIERA), Northwestern University, Evanston, IL 60208, USA Center for Interdisciplinary Exploration & Research in Astrophysics Texas Tech University, Lubbock, TX 79409, USA C 102 103 100 101 96 97 98 99 95 93 94 91 92 89 90 88 85 86 87 82 83 84 80 81 78 79 75 76 77 73 74 70 71 72 68 69 67 Ann. 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